# A model structure via orbit spaces for equivariant homotopy

## TL;DR

This paper constructs a new model structure on the category of right G-simplicial sets based on orbit spaces, providing a criterion for weak equivalences related to subgroup orbits in equivariant homotopy theory.

## Contribution

It introduces a left induced model structure on G-simplicial sets using orbit-based weak equivalences and cofibrations, advancing the understanding of equivariant homotopy models.

## Key findings

- Established a model structure where weak equivalences are detected on H-orbits for all H in a subgroup collection.
- Provided a categorical criterion linking orbit-wise weak equivalences to the overall model structure.
- Enhanced tools for analyzing equivariant homotopy types via orbit space models.

## Abstract

For a given group $G$ and a collection of subgroups $\mathcal F$ of $G$, we show that there exist a left induced model structure on the category of right $G$-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on the $H$-orbits for all $H$ in $\mathcal F$. This gives a model categorical criterion for maps that induce weak equivalences on $H$-orbits to be weak equivalences in the $\mathcal F$-model structure.